Combinatorics (math.CO)

Abstract: L. Soukup formulated an abstract framework in his introductory paper for proving theorems about uncountable graphs by subdividing them by an increasing continuous chain of elementary submodels. The applicability of this method relies on the preservation of a certain property (that varies from problem to problem) by the subgraphs obtained by subdividing the graph by an elementary submodel. He calls the properties that are preserved ``well-reflecting''. The aim of this paper is to investigate the possibility of weakening of the assumption ``well-reflecting'' in L. Soukup's framework. Our motivation is to gain better understanding about a class of problems in infinite graph theory where a weaker form of well-reflection naturally occurs.

Abstract: In 1930, Tammes posed the problem of determining $\rho(r, n)$, the minimum over all sets of $n$ unit vectors in $\mathbb{R}^r$ of their maximum pairwise inner product. In 1955, Rankin determined $\rho(r,n)$ whenever $n \leq 2r$ and in this paper we show that $\rho(r, 2r + k ) \geq \frac{\left(\frac{8}{27}k + 1\right)^{1/3} - 1}{2r + k}$, answering a question of Bukh and Cox. As a consequence, we conclude that the maximum size of a binary code with block length $r$ and minimum Hamming distance $(r-j)/2$ is at most $(2 + o(1))r$ when $j = o(r^{1/3})$, resolving a conjecture of Tiet\"av\"ainen from 1980 in a strong form. Furthermore, using a recently discovered connection to binary codes, this yields an analogous result for set-coloring Ramsey numbers of triangles.

Abstract: In an orientation $O$ of the graph $G$, an arc $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, the Frank number is the minimum $k$ for which $G$ admits $k$ strongly connected orientations such that for every edge $e$ of $G$ the corresponding arc is deletable in at least one of the $k$ orientations. H\"orsch and Szigeti conjectured the Frank number is at most $3$ for every $3$-edge-connected graph $G$. We prove an upper bound of $5$, which improves the previous bound of $7$.

Abstract: We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.

Abstract: We make a systematic study of matroidal mixed Eulerian numbers which are certain intersection numbers in the matroid Chow ring generalizing the mixed Eulerian numbers introduced by Postnikov. These numbers are shown to be valuative and obey a log-concavity relation. We establish recursion formulas and use them to relate matroidal mixed Eulerian numbers to the characteristic and Tutte polynomials, reproving results of Huh-Katz and Berget-Spink-Tseng. Generalizing Postnikov, we show that these numbers are equal to certain weighted counts of binary trees. Lastly, we study these numbers for perfect matroid designs, proving that they generalize the remixed Eulerian numbers of Nadeau-Tewari.

Abstract: A set $\mathcal{G}$ of planar graphs on the same number $n$ of vertices is called simultaneously embeddable if there exists a set $P$ of $n$ points in the plane such that every graph $G \in \mathcal{G}$ admits a (crossing-free) straight-line embedding with vertices placed at points of $P$. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether for some $n$ there exists a set $\mathcal{G}$ consisting of two planar graphs on $n$ vertices that are not simultaneously embeddable. While this remains widely open, we give a short proof that for every $\varepsilon>0$ and sufficiently large $n$ there exists a collection of at most $(4+\varepsilon)\log_2(n)$ planar graphs on $n$ vertices which cannot be simultaneously embedded. This significantly improves the previous exponential bound of $O(n\cdot 4^{n/11})$ for the same problem which was recently established by Goenka, Semnani and Yip.

Abstract: The generalized wreath product of association schemes was introduced by R.~A.~Bailey in European Journal of Combinatorics 27 (2006) 428--435. It is known as a generalization of both wreath and direct products of association schemes. In this paper, we discuss the Terwilliger algebra of the generalized wreath product of commutative association schemes. I will describe its structure and its central primitive idempotents in terms of the parameters of each factors and their central primitive idempotents.